Dab solver - Parseval's theorem

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More generally, given an abelian [[topological group]] G with [[Pontryagin dual]] G^, Parseval's theorem says the Pontryagin–Fourier transform is a unitary operator between Hilbert spaces L²(G) and L²(G^) (with integration being against the appropriately scaled  measures on the two groups.) When G is the unit circle T, G^ is the integers and this is the case discussed above. When G is the real line R, G^ is also R and the unitary transform is the [[Fourier transform]] on the real line. When G is the [[cyclic group]] Z_n, again it is self-dual and the Pontryagin–Fourier transform is what is called [[discrete-time Fourier transform]] in applied contexts.

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In [[mathematics]], '''Parseval's theorem''' <ref>Parseval des Chênes, Marc-Antoine "Mémoire sur les séries et sur l'intégration complète d'une équation aux différences partielles linéaire du second ordre, à coefficients constants" presented before the Académie des Sciences (Paris) on 5 April 1799. This article was published in ''Mémoires présentés à l’Institut des Sciences, Lettres et Arts, par divers savans, et lus dans ses assemblées. Sciences, mathématiques et physiques. (Savans étrangers.)'', vol. 1, pages 638–648 (1806).</ref> usually refers to the result that the [[Fourier transform]] is [[Unitary operator|unitary]]; loosely, that the sum (or integral) of the square of a function is equal to the sum (or integral) of the square of its transform. It originates from a 1799 theorem about [[series (mathematics)|series]] by [[Marc-Antoine Parseval]], which was later applied to the [[Fourier series]]. It is also known as '''Rayleigh's energy theorem''', or '''Rayleigh's Identity''', after [[John William Strutt]], Lord Rayleigh.<ref>Rayleigh, J.W.S. (1889) "On the character of the complete radiation at a given temperature," ''Philosophical Magazine'', vol. 27, pages 460–469. Available on-line [http://books.google.com/books?id=izM9AAAAIAAJ&pg=PA268&lpg=PA268&source=bl&ots=5stf6mGwJG&sig=UeoeV2c4dEp9JmWUIanqMEhDMmU&hl=en&ei=QTv9SZKTJIvOMrrxjL0E&sa=X&oi=book_result&ct=result&resnum=3 here].</ref>

Although the term "Parseval's theorem" is often used to describe the unitarity of ''any'' Fourier transform, especially in [[physics]] and [[engineering]], the most general form of this property is more properly called the [[Plancherel theorem]].<ref>Plancherel, Michel (1910) "Contribution a l'etude de la representation d'une fonction arbitraire par les integrales définies," ''Rendiconti del Circolo Matematico di Palermo'', vol. 30, pages 298–335.</ref>

== Statement of Parseval's theorem ==

Suppose that ''A''(''x'') and ''B''(''x'') are two [[square integrable]] (with respect to the [[Lebesgue measure]]), complex-valued functions on '''R''' of period 2π with [[Fourier series]]

:<math>A(x)=\sum_{n=-\infty}^\infty a_ne^{inx}</math>
and 
:<math>B(x)=\sum_{n=-\infty}^\infty b_ne^{inx}</math>

respectively. Then

:<math>\sum_{n=-\infty}^\infty a_n\overline{b_n} = \frac{1}{2\pi} \int_{-\pi}^\pi A(x)\overline{B(x)} \, dx,</math>

where ''i'' is the [[imaginary unit]] and horizontal bars indicate [[complex conjugation]].

More generally, given an abelian [[topological group]] ''G'' with [[Pontryagin dual]] ''G^'', Parseval's theorem says the Pontryagin–Fourier transform is a unitary operator between Hilbert spaces ''L''2(''G'') and ''L''2(''G^'') (with integration being against the appropriately scaled <<link:0>> measures on the two groups.) When ''G'' is the unit circle '''T''', ''G^'' is the integers and this is the case discussed above. When ''G'' is the real line '''R''', '''G^''' is also '''R''' and the unitary transform is the [[Fourier transform]] on the real line. When ''G'' is the [[cyclic group]] '''Z'''n, again it is self-dual and the Pontryagin–Fourier transform is what is called [[discrete-time Fourier transform]] in applied contexts.

== Notation used in engineering and physics ==

In [[physics]] and [[engineering]], Parseval's theorem is often written as:

:<math>\int_{-\infty}^\infty | x(t) |^2 \, dt = \int_{-\infty}^\infty | X(f) |^2 \, df </math>

where <math>X(f) = \mathcal{F} \{ x(t) \}</math> represents the [[continuous Fourier transform]] (in normalized, unitary form) of ''x''(''t'') and ''f'' represents the frequency component (not [[angular frequency]]) of ''x''.

The interpretation of this form of the theorem is that the total [[Energy (signal processing)|energy]] contained in a waveform ''x''(''t'') summed across all of time ''t'' is equal to the total energy of the waveform's Fourier Transform ''X''(''f'') summed across all of its frequency components ''f''.

For [[discrete time]] [[signal (information theory)|signals]], the theorem becomes:

:<math> \sum_{n=-\infty}^\infty | x[n] |^2 = \frac{1}{2\pi} \int_{-\pi}^\pi | X(e^{i\phi}) |^2 d\phi </math>

where ''X'' is the [[discrete-time Fourier transform]] (DTFT) of ''x'' and Φ represents the [[angular frequency]] (in [[radian]]s per sample) of ''x''.

Alternatively, for the [[discrete Fourier transform]] (DFT), the relation becomes:

:<math> \sum_{n=0}^{N-1} | x[n] |^2 = \frac{1}{N} \sum_{k=0}^{N-1} | X[k] |^2 </math>

where ''X''[''k''] is the DFT of ''x''[''n''], both of length ''N''.

== See also ==
*[[Bessel's inequality]]
*[[Parseval's identity]]
*[[Plancherel's theorem]]
*[[Parseval–Gutzmer formula]]

== Notes ==
{{reflist}}

==References==
* [http://www-history.mcs.st-andrews.ac.uk/Mathematicians/Parseval.html Parseval], ''MacTutor History of Mathematics archive''.
* George B. Arfken and Hans J. Weber, ''Mathematical Methods for Physicists'' (Harcourt: San Diego, 2001).
* Hubert Kennedy, ''[http://home.att.net/~clairnorman/Eight_Mathematical.pdf Eight Mathematical Biographies]'' (Peremptory Publications: San Francisco, 2002).
* Alan V. Oppenheim and Ronald W. Schafer, ''Discrete-Time Signal Processing'' 2nd Edition (Prentice Hall:  Upper Saddle River, NJ, 1999) p 60.
* William McC. Siebert, ''Circuits, Signals, and Systems'' (MIT Press: Cambridge, MA, 1986), pp.&nbsp;410–411.
* David W. Kammler, ''A First Course in Fourier Analysis'' (Prentice–Hall, Inc., Upper Saddle River, NJ, 2000) p.&nbsp;74.

==External links==
* [http://mathworld.wolfram.com/ParsevalsTheorem.html Parseval's Theorem] on Mathworld

[[Category:Theorems in Fourier analysis]]
[[Category:Theorems in harmonic analysis]]

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Dab solver - Parseval's theorem

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